Carlo Severini (10 March 1872 – 11 May 1951) was an Italian mathematician: he was born in Arcevia (Province of Ancona) and died in Pesaro. Severini, independently from Dmitri Fyodorovich Egorov, proved and published earlier a proof of the theorem now known as Egorov's theorem.
Biography He graduated in Mathematics from the University of Bologna on November 30, 1897:According to the summary of his student file available from the (an electronic version of the archives of the University of Bologna).The content of this section is based on references and : this last one also refers that he was married and had several children, however without giving any other detail. the title of his "Laurea" thesis was "Sulla rappresentazione analitica delle funzioni arbitrarie di variabili reali".An English translation reads as "On the Analytic Representation of Arbitrary Functions of Real variables"; despite the similarities in the title and the same year of publication, the biographical sources do not say if the paper is somewhat related to his thesis. After obtaining his degree, he worked in Bologna as an assistant to the chair of Salvatore Pincherle until 1900.The 1897–1898 yearbook of the university already lists him between the assistant professors. From 1900 to 1906, he was a senior high school teacher, first teaching in the Institute of Technology of La Spezia and then in the lyceums of Foggia and of Turin;According to . then, in 1906 he became full professor of Infinitesimal Calculus at the University of Catania. He worked in Catania until 1918, then he went to the University of Genova, where he stayed until his retirement in 1942.
Work He authored more than 60 papers, mainly in the areas of real analysis, approximation theory and partial differential equations, according to . His main contributions belong to the following fields of mathematics:Only his most known results are described in the following sections: reviews his research in greater detail.
Approximation theory In this field, Severini proved a generalized version of the Weierstrass approximation theorem. Precisely, he extended the original result of Karl Weierstrass to the class of bounded locally integrable functions, which is a class including particular discontinuous functions as members.According to , the result is given in various papers, source perhaps being the most accessible of them.
Measure theory and integration Severini proved Egorov's theorem one year earlier than Dmitri EgorovEgorov's proof is given in the paper . in the paper , whose main theme is nevertheless the study of sequences of orthogonal functions and their properties.Moreover, according to , while acknowledging his own priority in the publication of the result, Severini was unwilling to disclose it publicly: it was Leonida Tonelli who, in the note , publicly credited him the priority for the first time.
Partial differential equations Severini proved an existence theorem for the Cauchy problem for the non linear hyperbolic partial differential equation of first order
assuming that the Cauchy data (defined in the bounded interval ) and that the function has Lipschitz continuous first order partial derivatives,This means that f belongs to the class . jointly with the obvious requirement that the set is contained in the domain of .For more details about his researches in this field, see and the references cited therein
Real analysis and unfinished works According to , he worked also on the foundations of the theory of real functions. lists Severini's researches on this field under as "Fondamenti dell'analisi infinitesimale (Foundations of infinitesimal analysis)": however, the topics covered range from the theory of integration to absolutely continuous functions and to operations on series of real functions. Severini also left an unpublished and unfinished treatise on the theory of real functions, whose title was planned to be "Fondamenti dell'analisi nel campo reale e i suoi sviluppi"."Foundations of Analysis on the Real Field and its Developments": again according to , the treatise would have included his later original results and covered all the fundamental topics required for the study of functional analysis on the real field.
Selected publications . In this paper Severini extends the standard Weierstrass approximation theorem to a wider class of functions characterised by the fact that they can have a particular kind of discontinuities. . This paper contains Severini's most known and cited result, i.e. the Severini–Egorov theorem.
See also Hyperbolic partial differential equation Orthogonal functions Severini-Egorov theorem Weierstrass approximation theorem
Notes
References
Biographical and general references . A very short summary of the student file of Carlo Severini, giving however useful information about his laurea. , available from the Biblioteca Digitale Italiana di Matematica. The obituary of Carlo Severini. . In this short note Leonida Tonelli credits Severini for the first proof of Severini–Egorov theorem. . "Italian mathematicians of the first century of the unitary state" is an important historical memoir giving brief biographies of the Italian mathematicians who worked and lived between 1861 and 1961. Its content is available from the website of the . .
Scientific references . A monograph surveying the theory of hyperbolic equations up to its state of the art in the early 1960s, published by the Consiglio Nazionale delle Ricerche. , available at Gallica.
External links . Available from the Edizione Nazionale Mathematica Italiana.